Note that, apparently, the right-hand side can only be a tau function of the hypergeometric type of the
TL hierarchy [1]},[2]},[3]},[4]}. It's not an exact statement,
but I don't see any other possibilities at the moment.
That is, there may be a tau function of the form
\(\sum _\lambda s_\lambda (\mathbf {p}^1)s_\lambda (\mathbf {p}^2)\prod _{(i,j)\in \lambda } r(j-i)\) 
